Probability of promoter occupancy

**(a)** Schematic showing how, in the simple model, the DNA molecule serves as a reservoir for the RNAP molecules, almost all of which are bound to DNA.

**(b)** Illustration of the states of the promoter – either with RNAP not bound or bound and the remaining polymerase molecules distributed among the non-specific sites. The statistical weights associated with these different states of promoter occupancy are also shown.

**(c)** Probability of binding of RNAP to promoter as a function of the number of RNAP molecules for two different promoters. We assume the number of non-specific sites is

*N*_{NS} = 5 × 10

^{6}, and calculate the binding energy difference using the simple relation

, where the equilibrium dissociation constants for specific binding (

) and non-specific binding (

) are taken from in vitro measurements. In particular, making the simplest assumption that the genomic background for RNAP is given only by the non-specific binding of RNAP with DNA, we take

[], for the lac promoter

[] and for the T7 promoter,

[]. For the lac promoter, this results in Δ

*ε*_{pd} = −2.9

*k*_{B}T and for the T7 promoter, Δ

*ε*_{pd} = −8.1

*k*_{B}T.

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